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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 113
Chapter 3, Problem 113

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.
(g○ƒ)(x)(g ○ ƒ)(x)

Verified step by step guidance
1
Understand that the composition of functions (g \(\circ\) ƒ)(x) means you substitute the output of ƒ(x) into g(x). In other words, (g \(\circ\) ƒ)(x) = g(ƒ(x)).
Identify the given functions: ƒ(x) = \(\sqrt{x - 2}\) and g(x) = x^2.
Substitute ƒ(x) into g(x): replace every x in g(x) with \(\sqrt{x - 2}\), so g(ƒ(x)) = (\(\sqrt{x - 2}\))^2.
Simplify the expression (\(\sqrt{x - 2}\))^2 by using the property that squaring a square root returns the original expression inside the root, so it simplifies to x - 2.
Determine the domain of the composition (g \(\circ\) ƒ)(x) by considering the domain of ƒ(x), which requires x - 2 \(\geq\) 0, so x \(\geq\) 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Composition

Function composition involves applying one function to the result of another, denoted as (g ○ f)(x) = g(f(x)). It requires substituting the entire output of the inner function into the outer function. Understanding this process is essential to correctly evaluate composite functions.
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Function Composition

Domain of a Function

The domain is the set of all input values for which a function is defined. When composing functions, the domain of the composite function depends on the domain of the inner function and the domain restrictions of the outer function after substitution. Identifying these restrictions ensures valid inputs.
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Domain Restrictions of Composed Functions

Square Root and Quadratic Functions

The function ƒ(x) = √(x-2) is a square root function, defined only for x ≥ 2 to keep the expression under the root non-negative. The function g(x) = x² is a quadratic function, defined for all real numbers. Knowing their properties helps in evaluating and composing these functions correctly.
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Solving Quadratic Equations by the Square Root Property
Related Practice
Textbook Question

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.

(f+g)(2k)(f+g)(2k)

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible

(g○ƒ)(3) (g ○ ƒ)(3)

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Textbook Question

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.

(f/g)(−1)(f/g)(-1)

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

(f○g)(−6)(f ○ g)(-6)

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Textbook Question

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

(ƒ○g)(x)(ƒ ○ g)(x)

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